Normal Mode Method Research Articles

Effects of rotation and chemical reactions on an electroconvection in Maxwell dielectric nanofluids within anisotropic porous media

ABSTRACT This study explores the complex interactions between rotation, chemical reactions, and electroconvection in Maxwell dielectric nanofluids within anisotropic porous media. The research aims to elucidate how these factors influence the heat transfer efficiency and fluid dynamics under the influence of external alternating current (AC) electric fields. The normal mode method is employed to analyse the stability of the system, allowing for the examination of oscillatory modes and their growth rates. By introducing perturbations in the physical parameters, the study derives a set of ordinary linear differential equations that characterize the system's response to external influences. The coupled differential equations were solved using the Galerkin approach for first approximation while meeting the necessary boundary conditions. Analytical solution of the representative equation for stress-free boundary states yields Rayleigh number formulas for nonoscillatory and oscillatory mode occur. Oscillatory modes are seen for both bottom and top-heavy nanoparticle distributions. The electric Rayleigh number, thermal Prandtl number, and stress relaxation parameter increases, whereas the Brinkman-Darcy number delays the onset of stationary and oscillatory convection. The study finds that positive concentration Rayleigh numbers stabilize the system, while negative ones lead to instability. Additionally, it demonstrates that external alternating current electric fields influence convection behavior and provides formulas for critical Rayleigh numbers, enhancing our understanding of fluid interactions in various engineering and technological applications.

Optoelectronic and photoacoustic waves in rotating excited semiconductor subjected to thermal shock

In this study, the impact of rotation on the transmission of optical acoustic waves, arising from the movement of the optical carrier within an elastic thermal environment, is examined through theoretical analysis. The theoretical framework incorporates the interaction between acoustic waves and thermomechanics to formulate governing equations specific to a semiconductor medium. The fundamental equations of the physical model, influenced by photothermal and thermoelastic principles, are deduced mathematically while considering the presence of rotation. The model studied the medium under a thermal shock wave from thermal stress resulting from light-induced temperature elevation that is not fixed but decaying over time. The mathematical model is solvable using the normal mode method. The model’s numerical solutions provide access to all physical fields within the physical domain, encompassing displacement, temperature, acoustic pressure, mechanical stress, and carrier density diffusion. Utilizing the harmonic wave method, a two-dimensional graphical representation of the rotation parameter with and without the influence of the decay parameter is obtained. The theoretical analysis involves comparison, analysis, and discussion of the effects of these parameters investigated.

Theoretical investigation of the excitation of leaky modes for multilayered models

SUMMARY The dispersion analysis with array-based methods enables us to detect and analyse the multimode dispersive waves with weak amplitudes. Over the past years, there has been widespread extraction of higher normal modes from earthquake records and ambient noise, which have been extensively employed in near-surface and crustal investigations. Notably, the frequent reporting of leaky modes in recent observations has gained significant attention. This is because leaky modes can provide more constraints on subsurface structures. Particularly, a subset of leaky modes is found to be sensitive to P-wave velocities, which has the potential to compensate for the limitations of traditional surface wave inversion. Nevertheless, due to the lack of knowledge of the excitation and practical application of leaky modes, it is urgent to have an effective guide in selecting certain leaky modes for practical inversion combined with other imaging methods. To this end, the theoretical investigation into the excitation of leaky modes for multilayered models is undertaken. By computing theoretical seismograms for various source mechanisms and source depths with the discrete wavenumber and the normal-mode summation methods, respectively, the qualitative evaluation of the leaky-mode contributions to the synthesized seismograms is conducted. Additionally, the impact of the source depth for the same source mechanism is meticulously examined by the dispersion spectra. Furthermore, with the accurate computation of leaky modes, we categorize the leaky modes into PL and OP (organ-pipe) modes according to their distinctive properties, that is, their attenuation and eigen-displacements, which may be used to explain very well the occurrence of certain leaky modes on the dispersion spectrum and be indicative of the reasonable choice of PL modes to put them into practical inversion for P-velocity structures.

Weakly nonlinear analysis of rotating magnetoconvection with anisotropic thermal diffusivity effect

The influence of anisotropic thermal diffusive coefficient on the stability of the horizontal fluid planar layer rotating about its vertical axis and permeated by the horizontal homogeneous magnetic field is studied. The linear stability analysis is carried out using the normal mode method. The stationary cross/oblique and parallel modes are calculated for different ranges of control parameters arising in the system. The SA (Stratification Anisotropy) parameter, α (the ratio of horizontal and vertical thermal diffusivities), plays a key role in deciding the boundaries between these modes and their instability regions when there is a combination of high and low rotation with weak and strong magnetic fields. The obtained isotropic results coincide with those obtained by pioneers in the literature. The weakly nonlinear behavior of the stationary convective motion in the vicinity of primary instability threshold is studied using the two-dimensional Landau–Ginzburg (LG) equation with cubic nonlinearity. This equation derived using the multiple scale analysis is similar to the one obtained in the literature having different relaxation time, nonlinear coefficient, and coherence lengths. These coefficients are used to study the heat transfer rate. In the case of high rotation, Nusselt number gets decreased from atmospheric (α < 1) to oceanic (α > 1) SA types. The domain for secondary instability of Eckhaus is obtained using the spatiotemporal LG equation and it is observed that the Eckhaus instability region decreases with increasing α.

Analytical insight into local defect resonance using a two-step method

Local defect resonance (LDR) has gained increasing attention in the field of non-destructive testing due to its capacity to selectively amplify vibrations within the defect regions and to enable defect-selective and high-resolution imaging. How-ever, the conventional theory based on vibration method adopted for analyzing the fundamental frequencies of LDR deviates from experimental results when dealing with relatively thick defects. To tackle this deficiency, this study presents a two-step analytical methodology designed to precisely evaluate the local reso-nance effects of thick planar defects. In the first step, the normal-mode expansion method is employed to calculate the phase shift of the reflected elastic wave. In the second step, by using the standing wave's wavelength obtained in the first step, the Rayleigh method is taken into account to calculate LDR frequencies for various defect shapes. The refined theory is subsequently validated through finite element analysis and experimental measurement. The results clearly demonstrate that the LDR frequencies calculated using the refined theory exhibit a better agreement with experimental and simulated data.

Моделирование КВ-радиотрасс на основе волноводного подхода

We present a scheme for modeling HF radio signal characteristics along paths of different lengths, which is based on the waveguide approach — the normal mode method. We use a representation of the recorded signal field in the form of Green function products of the angular operator, excitation coefficients, and reception coefficients of individual normal modes. Algorithms have been developed for calculating distance-frequency, frequency-angular, and amplitude characteristics of signals in large spatial regions through analysis and numerical summation of normal mode series. We have implemented a complex algorithm for simulating propagation conditions of HF radio signals, which includes a medium model, algorithms for calculating signal characteristics, and operational diagnostics of radio channel. We have compared the results of the HF signal propagation characteristic modeling and the experimental oblique sounding data obtained along paths of different lengths and orientation. To analyze experimental ionograms, determine the maximum usable frequencies for propagation modes along radio paths, we employ the method of automatic processing and interpretation of oblique sounding ionograms.

HF radio path modeling by waveguide approach

We present a scheme for modeling HF radio signal characteristics along paths of different lengths, which is based on the waveguide approach — the normal mode method. We use a representation of the recorded signal field in the form of Green function products of the angular operator, excitation coefficients, and reception coefficients of individual normal modes. Algorithms have been developed for calculating distance-frequency, frequency-angular, and amplitude characteristics of signals in large spatial regions through analysis and numerical summation of normal mode series. We have implemented a complex algorithm for simulating propagation conditions of HF radio signals, which includes a medium model, algorithms for calculating signal characteristics, and operational diagnostics of radio channel. We have compared the results of the HF signal propagation characteristic modeling and the experimental oblique sounding data obtained along paths of different lengths and orientation. To analyze experimental ionograms, determine the maximum usable frequencies for propagation modes along radio paths, we employ the method of automatic processing and interpretation of oblique sounding ionograms.

Transient behavior of a plate partially immersed in the fluid subjected to impact loadings: Theoretical analysis and experimental measurements

This study aimed to investigate the transient behavior of a rectangular plate partially in contact with fluid subjected to a dynamic external force. To this purpose, a theoretical model was developed to analyze vibration characteristics and transient wave propagation. Based on superposition method, the dry mode shapes and natural frequencies of the plate under vacuum could be obtained. The dry mode shapes were treated as the fundamental function to construct the wet mode that describes the vibration behavior of the fluid-plate coupled system. The velocity potential and fluid pressure within a finite tank due to the plate deflection were derived using an equation governing the incompressible fluid. Based on the relationship between dry mode shape and wet mode shape, the fluid-plate coupled system's wet mode shape and resonant frequency could be determined from the frequency response function. Applying the normal mode method, the transient displacement of plate and fluid pressure can be obtained by solving a system of non-homogeneous differential equations. The theoretical predictions were verified by finite element method (FEM) and experimental measurements. Experiments were conducted using piezoelectric film sensors (polyvinylidene fluoride, PVDF) to measure the force history induced by a steel ball impact to quantitatively analyze the transient response. The comparison results proved that the theoretical predictions and experiments were in good agreement, including the transient responses of the displacement and in-plane strain of a plate partially submerged in the fluid. The results indicate that changes in water depth can induce resonance frequency shifts and wet mode shape distortions, which also illustrate that the vibrational properties of wet modes affect transient behavior. The proposed transient solution demonstrates an analytical approach that connects the physical significance of the dynamic behavior of the fluid-plate coupled system in time and frequency domains; it provides a connection between the transient behaviors and vibration characteristics.

Onset of rotating convection of a nanofluid with helical force: Stability analysis

Employing linear stability analysis of nanofluids holds promising advantages across diverse domains. The impact of rotation and helical force on the onset of convection in the nanofluid is investigated using the linear stability analysis. The method of normal modes is used to solve the governing nondimensional equations, which results in an eigenvalue problem for the linear stability analysis. The bvp4c in MATLAB R2021a is employed to solve the eigenvalue problem. Three different lower-upper boundary conditions are considered: free-free, rigid-free, and rigid-rigid. The Rayleigh and wave number are determined for various required values of the other physical parameters and graphically displayed. Specifically, as the Taylor number rises, there is a corresponding grow in the critical Rayleigh number. This indicates that higher values of the Taylor number stabilize the system, making it less prone to convective instabilities. Conversely, the Lewis number and Helical force parameter destabilize the critical Rayleigh number, making the system more prone to convective instabilities across all boundaries. This information provides valuable insights into the system’s dynamics under study and can be crucial for predicting and understanding its behavior under different conditions.

Tropical and Subtropical South American Intraseasonal Variability: A Normal-Mode Approach

Instead of using the traditional space-time Fourier analysis of filtered specific atmospheric fields, a normal-mode decomposition method was used to analyze South American intraseasonal variability (ISV). Intraseasonal variability was examined separately in the 30–90-day band, 20–30-day band, and 10–20-day band. The most characteristic structure in the intraseasonal time-scale, in the three bands, was the dipole-like convection between the South Atlantic Convergence Zone (SACZ) and the central-east South America (CESA) region. In the 30–90-day band, the convective and circulation patterns were modulated by the large-scale Madden–Julian oscillation (MJO). In the 20–30-day and 10–20-day bands, the convection structures were primarily controlled by extratropical Rossby wave trains. The normal-mode decomposition of reanalysis data based on 30–90-day, 20–30-day, and 10–20-day ISV showed that the tropospheric circulation and CESA–SACZ convective structure observed over South America were dominated by rotational modes (i.e., Rossby waves, mixed Rossby-gravity waves). A considerable portion of the 30–90-day ISV was also associated with the inertio-gravity (IGW) modes (e.g., Kelvin waves), mainly prevailing during the austral rainy season. The proposed decomposition methodology demonstrated that a realistic circulation can be reproduced, giving a powerful tool for diagnosing and studying the dynamics of waves and the interactions between them in terms of their ability to provide causal accounts of the features seen in observations.

Proper generalized decomposition-based iterative enrichment process combined with shooting method for steady-state forced response analysis of nonlinear dynamical systems

This paper presents a novel framework combining proper generalized decomposition (PGD) with the shooting method to determine the steady-state response of nonlinear dynamical systems upon a general periodic input. The proposed PGD approximates the response as a low-rank separated representation of the spatial and temporal dimensions. The Galerkin projection is employed to formulate the subproblem for each dimension, then the fixed-point iteration is applied. The subproblem for the spatial vector can be regarded as computing a set of reduced-order basis vectors, and the shooting problem projected onto the subspace spanned by these basis vectors is defined to obtain the temporal coefficients. From this procedure, the proposed framework replaces the complex nonlinear time integration of the full-order model with the series of solving simple iterative subproblems. The proposed framework is validated through two descriptive numerical examples considering the conventional linear normal mode method for comparison. The results show that the proposed shooting method based on PGD can accurately capture nonlinear characteristics within 10 modes, whereas linear modes cannot easily approximate these behaviors. In terms of computational efficiency, the proposed method enables CPU time savings of about one order of magnitude compared with the conventional shooting methods.

A new model of rotating nonlocal fiber-reinforced visco-thermoelastic solid using a modified Green–Lindsay theory

The effect of rotation on a nonlocal fiber-reinforced visco-thermoelastic media was examined in this work using an modified Green and Lindsay theory (MGL). The problem was resolved by using normal mode method to derive the precise expressions of field quantities. In this technique, one gets exact solution without any assumed restrictions on the field variables. The normal mode technique is applicable to a wide range of problems in thermodynamics and thermoelasticity. Graphical representations of the thermal temperature, displacements and stresses are obtained. Comparisons of the physical quantities are shown in figures to study the effects of nonlocal parameter, rotation, viscosity and reinforcement parameters. Some special cases of interest have also been inferred from the present problem. The results indicate that rotation, nonlocal parameter, viscosity and reinforcing factors have a considerable impact on the fluctuations of the variables under consideration. These impacts are examined and described in depth.

Study of the strength characteristics of particle velocity at the seabed interface

When studying the propagation characteristics of the acoustic vector field, the law that the horizontal particle velocity intensity is stronger than the vertical particle velocity intensity does not apply near the seabed interface. To verify this characteristic, a sea experiment was conducted in the South China Sea to obtain the long-range propagation data of particle velocity at the seabed interface. Then, based on the normal mode method, the particle velocity intensity distribution of different types of seabed is theoretically simulated and compared with the experimental data, and the intensity characteristics of particle velocity at the seabed interface are verified. The results show that the particle velocity strength of the elastic seabed is mainly influenced by the submarine shear wave, which leads to the phenomenon that the vertical particle velocity is stronger than the horizontal particle velocity strength, and the relative strength of the two is determined by the compression wave velocity, the shear wave velocity and the seabed density.

STUDY OF GLOBAL STABILITY OF ROTATING PARTIALLY IONIZED PLASMA SATURATING A POROUS MEDIUM

The importance of thermal convection in rotating partially ionized plasma has been observed in various laboratory and astrophysical plasmas. The focus of this work is on the investigation of the effect of rotation on the thermal convection of partially ionized plasma within a porous medium by using nonlinear and linear analyzes. For porous medium, the Darcy-Brinkman model has been used. The eigenvalue problems for linear and nonlinear analyzes have been developed using the normal mode method and energy method, respectively. For numerical analysis, the Galerkin-weighted residual method has been employed to determine the Rayleigh-Darcy number. The effects of rotation, medium permeability, compressibility, and collisional frequency have been observed on the stability of the system. It has been found that the subcritical region does not exist, and hence global stability prevails. The rotation is found to induce oscillatory modes of convection. Rotation, medium permeability, and compressibility are found to delay the onset of convection. The collisional frequency doesn't influence the stability of the system for stationary convection; however, it does influence energy decay and oscillatory convection. All the findings of our study have been discussed and presented graphically.

The Onset of Instability in A Magnetohydrodynamic Channel Flow through Porous Media of Casson Fluid

A detailed study is made on the stability of linear two-dimensional disturbances of Plane Poiseuille Flow (PPF) of Casson fluid through porous media in the presence of a vertical uniform magnetic field, B0 which is extremely useful in metals, mines, and fuels industries. Using the method of normal modes, the disturbance equations are derived. The resulting eigenvalue problem is then solved by the spectral collocation method using Chebyshev-based polynomials. The critical values of the triplets ( Rec, αc, cc ) are obtained for various values of the Casson parameter, η , Hartmann number, Ha , and porous parameter, σp. The stability of the system is discussed using the neutral stability curves for each value of the parameters present in the problem. It is found that the stability regions are enlarged for small values of η and large values of the porous parameter, σp and Hartmann number, Ha. It is also observed that the stability characteristics of plane Poiseuille flow in a porous medium are remarkably different from non-porous cases. The results obtained here contribute to the contemporary efforts to better understand the stability characteristics of PPF of Casson fluid flow through porous media in the presence of a uniform transverse magnetic field.

Simulation result of the effects of radiation noise from long-distance ships on marine mammals

In recent, there is a worldwide concern that underwater noise from human activities may have impacted marine life. The international community recognizes the underwater radiated noise from navigational ships may have consequences on marine life. However, there is no scientific evidence for these, so they need to be investigated. MLIT launched the project with the aim of understanding the current status of underwater noise by navigational ships. In this project, a hydrophone has been installed in the sea off Ohshima Island since 2020 to measure the noise by navigational ships around the hydrophone. The purpose of this study , first is to calculate the source level of ship radiation noise using the normal mode method with the ship noise data obtained from hydrophone measurements. Second is to simulate the propagation of the source level using Parabolic Equation (PE) with the marine environment of the propagation path as a parameter. Finally, based on the results of propagation simulations, we examine the impact of ship noise on whales. As a result, revealed the noise levels emitted from RORO vessels, and by simulating the propagation of the noise levels, it was possible to confirm the distances that affect the whale's hearing.

Jeans Gravitational Instability of a Rotating Collisionless Magnetized Plasma with Anisotropic Pressure

The problem of self-gravitational instability of an astrophysical rotating plasma in a strong magnetic field with an anisotropic pressure tensor is studied on the basis of the Chew–Goldberger–Low (CGL) quasi-hydrodynamic equations modified by generalized polytropic laws. Using the general form of a dispersion relation obtained by the normal-mode perturbation method, a discussion is provided of the propagation of small-amplitude perturbation waves in an infinite homogeneous plasma medium for transverse, longitudinal, and oblique directions with respect to the magnetic field vector. It is shown that different polytropic indices and anisotropic pressures not only change the classical Jeans instability condition but also cause the appearance of new unstable regions. Modified Jeans instability criteria are obtained for isotropic MHD equations and anisotropic CGL equations owing to the influence of the polytropic indices on gravitational and firehose instabilities for astrophysical plasma. It is shown that in the case of a longitudinal mode of perturbation wave propagation, the Jeans instability criterion does not depend on uniform rotation. In the case of the transverse propagation regime, the presence of rotation reduces the critical wave number and exerts a stabilizing effect on the growth rate of the unstable regime.

Effects of Neumann and Robin boundaries on the thermal instability

AbstractThe thermo convective instability of the Darcy-Benard problem (DB) using Robin (third-kind) thermal conditions is investigated here. We consider a viscous Newtonian fluid saturating a porous layer in which the layer is sandwiched between two impermeable boundaries. The upper and the lower walls are modelled in the form of the Neumann (second-kind) and the Robin (third-kind) thermal conditions, respectively. The difference in the temperature distribution between both phases allows the lack of a local thermal equilibrium model to be present. As a consequence, the third kind of thermal condition brings about one extra dimensionless parameter of the Biot number to the usual one of the inter-heat transfer coefficient and the thermal conductivity ratio. The normal modes method adopted in a linear stability analysis gives rise to perturbed governing equations. The eigenvalue problem is handled numerically as a result of the perturbed governing equations leading to the marginal stability condition.

The Combined Effect of Gravity Modulation and Throughflow on Thermal Instability in the Hele-Shaw Cell Filled with Oldroyd-B Nanofluid

This paper shows the combined effect of throughflow and gravity modulation on the stability of Oldroyd-B nanofluid filled in Hele-Shaw cell. Nanofluid compared to the base fluid has higher thermal conduction. The thermal conductivity of nanofluid increased and thus increases the amount of energy transferred. The Oldroyd-B fluid model is important because of its numerous applications such as production of plastic sheet and extrusion of polymers through a slit die in polymer industry, biological solution pant tars glues, etc. In linear stability analysis, we found the expression of the critical Hele-Shaw Rayleigh number by using the normal mode method. Two-term Fourier series method is used for non-linear stability analysis and is also considered the Brinkman model for flow of nanofluid in Hele-Shaw cell. In linear stability analysis, we observed that there is no effect of Oldroyd-B nanofluid, which means that Deborah number (λ1) and retardation parameter (λ2) do not affect the stability analysis. Oldroyd-B nanofluid is similar to ordinary nanofluid in linear analysis. In non-linear analysis, Deborah number, retardation parameter, throughflow, gravity modulation, and Hele-Shaw number play a major role in heat/mass transfer. Enhancement in both heat/mass transfer in the system while increasing throughflow and Deborah number. An increment in Hele-Shaw number (Hs), decreases heat/mass transfer in the system.

Integral equations of the second kind in dynamic analysis of nonlinear systems with a finite number of degrees of freedom under arbitrary dynamic loading and material dependencies

Introduction. Development of computational methods for nonlinear systems possesses a significant potential, considering that linear theory sometimes fails to accurately describe the properties of dynamic systems, and the linear approximation gives only a very rough idea of real processes for a number of cases.Aim. Calculating linear systems and deriving the resulting equations for nonlinear systems involves impulse response and transfer functions of linear “generating” systems of differential equations. Such an approach in comparison with the traditional method of the so-called normal forms enables the calculation algorithm to be simplified considerably, avoiding several stages and presenting the solution by means of the normal mode method for linear systems directly with respect to the generalized coordinates.Materials and methods. The paper presents a method and an algorithm developed for the calculation of nonlinear systems with a finite number of degrees of freedom under arbitrary dynamic loading and material nonlinearity. Systems of nonlinear differential equations were reduced to nonlinear integral equations of the second kind, considered as resulting equations. The solution was developed in time steps, the value of which, among other things, determines the accuracy of the solution and the nature of the computational algorithm.Results. The paper presents main computational dependencies in a generalized form, convenient for numerical simulation. The author provides solutions for a nonlinear system with one degree of freedom and a cubic reactiondisplacement relation, as well as for a system with one and two degrees of freedom with a viscous damper. In both cases, the developed solution contains all properties of nonlinear systems, including the jump (transition) from the upper ascending branch to the lower, stable one, and the associated excitation of free oscillations.Conclusions. According to the calculations, the occurrence of nonlinear effects in oscillating systems makes positive impact on their behavior, in resonant modes in particular.